Started adding markdown

.gitignore

@@ -1 +1,5 @@
 *.agdai
+out/
+*.pdf
+*.log
+Everything.agda

ElgotAlgebra.lagda.md

@@ -1,7 +1,6 @@
+<!--
+```agda
 open import Level
-
-module ElgotAlgebra where
-
 open import Categories.Functor renaming (id to idF)
 open import Categories.Functor.Algebra
 open import Categories.Category
@@ -11,6 +10,20 @@ open import Categories.Category.Cocartesian
 open import Categories.Category.Extensive.Bundle
 open import Categories.Category.Extensive
 import Categories.Morphism.Reasoning as MR
+```
+-->
+
+## Summary
+This file introduces (guarded) elgot algebras
+
+- [X] *Definition 7* Guarded Elgot Algebras
+- [ ] *Theorem 8* Existence of final coalgebras is equivalent to existence of free H-guarded Elgot algebras
+- [X] *Proposition 10* Characterization of unguarded elgot algebras
+
+## Code
+
+```agda
+module ElgotAlgebra where
 
 private
   variable
@@ -22,10 +35,10 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
   open Cocartesian (Extensive.cocartesian extensive)
   open Cartesian (ExtensiveDistributiveCategory.cartesian D)
   open MR C
+```
 
-  --*
-  -- F-guarded Elgot Algebra
-  --*
+### *Definition 7* Guarded Elgot Algebras
+```agda
   module _ {F : Endofunctor C} (FA : F-Algebra F) where
     record Guarded-Elgot-Algebra : Set (o ⊔ ℓ ⊔ e) where
       open Functor F public
@@ -47,10 +60,13 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
           → f ≈ g 
           → (f #) ≈ (g #)
 
+```
 
-  --*
-  -- (unguarded) Elgot-Algebra
-  --*
+### *Proposition 10* Unguarded Elgot Algebras
+Unguarded elgot algebras are `Id`-guarded elgot algebras.
+Here we give a different Characterization and show that it is equal.
+
+```agda
   record Elgot-Algebra-on (A : Obj) : Set (o ⊔ ℓ ⊔ e) where
     -- iteration operator
     field
@@ -231,3 +247,4 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
       [ idC , idC ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] # ] 
       ∘ ([ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ i₂)                  ≈˘⟨ pushˡ #-Fixpoint ⟩ 
       [ (idC +₁ i₁) ∘ f , i₂ ∘ h ] # ∘ i₂                    ∎
+```

Makefile

@@ -0,0 +1,17 @@
+.PHONY: all clean
+
+all: MonadK.lagda.md
+	agda --html --html-dir=out MonadK.lagda.md --html-highlight=auto
+	agda --html --html-dir=out ElgotAlgebra.lagda.md --html-highlight=auto
+	pandoc out/MonadK.md -s -c Agda.css -o out/MonadK.html
+	pandoc out/ElgotAlgebra.md -s -c Agda.css -o out/ElgotAlgebra.html
+
+clean:
+	rm -rf out/*
+
+open:
+	firefox out/MonadK.html
+	firefox out/ElgotAlgebra.html
+
+Everything.agda:
+	git ls-tree --full-tree -r --name-only HEAD | grep '^[^\.]*.agda' | sed -e 's|^[/]*|import |' -e 's|/|.|g' -e 's/.agda//' -e '/import Everything/d' | LC_COLLATE='C' sort > Everything.agda

Monad/Instance/Delay.lagda.md

@@ -1,3 +1,20 @@
+---
+title: Delay Monad
+author: Leon Vatthauer
+format: pdf
+output:
+  pdf_document:
+    md_extensions: +task-lists
+mainfont: DejaVu Serif
+monofont: mononoki
+geometry: margin=0.5cm
+header-includes:
+ - \usepackage{fvextra}
+ - \DefineVerbatimEnvironment{Highlighting}{Verbatim}{breaklines,commandchars=\\\{\}}
+---
+
+<!--
+```agda
 open import Level
 open import Categories.Category.Core
 open import Categories.Category.Distributive
@@ -13,7 +30,10 @@ open import Categories.Functor
 open import Categories.Monad.Construction.Kleisli
 import Categories.Morphism as M
 import Categories.Morphism.Reasoning as MR
+```
+-->
 
+```agda
 module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ e) where
   open ExtensiveDistributiveCategory ED renaming (U to C; id to idC)
   open Cocartesian (Extensive.cocartesian extensive)
@@ -66,3 +86,6 @@ module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ
       ; sym-assoc = λ {X} {Y} {Z} {f} {g} → *-unique ((g *) ∘ f) ((g *) ∘ (f *))
       ; extend-≈ = *-resp-≈
       }
+
+  -- record Search
+```

MonadK.agda

@@ -1,40 +0,0 @@
-open import Level
-open import Categories.Category.Core
-open import Categories.Category.Extensive.Bundle
-open import Function using (id)
-
-module MonadK {o ℓ e} (D : ExtensiveDistributiveCategory o ℓ e) where
-  open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
-  open import UniformIterationAlgebras
-  open import UniformIterationAlgebra
-  open import Categories.FreeObjects.Free
-  open import Categories.Functor.Core
-  open import Categories.Adjoint
-  open import Categories.Adjoint.Properties
-  open import Categories.NaturalTransformation.Core renaming (id to idN)
-  open import Categories.Monad
-  open Equiv
-
-  record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
-    forgetfulF : Functor (Uniform-Iteration-Algebras D) C
-    forgetfulF = record
-      { F₀ = λ X → Uniform-Iteration-Algebra.A X
-      ; F₁ = λ f → Uniform-Iteration-Algebra-Morphism.h f
-      ; identity = refl
-      ; homomorphism = refl
-      ; F-resp-≈ = id
-      }
-    
-    field
-      algebras : ∀ X → FreeObject {C = C} {D = Uniform-Iteration-Algebras D} forgetfulF X
-
-    freeF : Functor C (Uniform-Iteration-Algebras D)
-    freeF = FO⇒Functor forgetfulF algebras
-    
-    adjoint : freeF ⊣ forgetfulF
-    adjoint = FO⇒LAdj forgetfulF algebras
-
-    K : Monad C
-    K = adjoint⇒monad adjoint
-
-    -- TODO show that the category of K-Algebras is the category of uniform-iteration algebras

MonadK.lagda.md

@@ -0,0 +1,76 @@
+<!--
+```agda
+open import Level
+open import Categories.Category.Core
+open import Categories.Category.Equivalence using (StrongEquivalence)
+open import Categories.Category.Extensive.Bundle
+open import Function using (id)
+open import UniformIterationAlgebras
+open import UniformIterationAlgebra
+open import Categories.FreeObjects.Free
+open import Categories.Functor.Core
+open import Categories.Adjoint
+open import Categories.Adjoint.Properties
+open import Categories.Adjoint.Monadic.Crude
+open import Categories.NaturalTransformation.Core renaming (id to idN)
+open import Categories.Monad
+open import Categories.Category.Construction.EilenbergMoore
+open import Categories.Category.Slice
+```
+-->
+
+## Summary
+In this file I explore the monad ***K*** and its properties:
+
+- [X] *Lemma 16* Definition of the monad
+- [ ] *Lemma 16* EilenbergMoore⇒UniformIterationAlgebras (use [crude monadicity theorem](https://agda.github.io/agda-categories/Categories.Adjoint.Monadic.Crude.html))
+- [ ] *Proposition 19* ***K*** is strong
+- [ ] *Theorem 22* ***K*** is an equational lifting monad
+- [ ] *Proposition 23* The Kleisli category of ***K*** is enriched over pointed partial orders and strict monotone maps
+- [ ] *Proposition 25* ***K*** is copyable and weakly discardable
+- [ ] *Theorem 29* ***K*** is an initial pre-Elgot monad and an initial strong pre-Elgot monad
+
+
+## Code
+
+```agda
+module MonadK {o ℓ e} (D : ExtensiveDistributiveCategory o ℓ e) where
+  open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
+  open Equiv
+
+  -- TODO move this to a different file
+
+  forgetfulF : Functor (Uniform-Iteration-Algebras D) C
+  forgetfulF = record
+    { F₀ = λ X → Uniform-Iteration-Algebra.A X
+    ; F₁ = λ f → Uniform-Iteration-Algebra-Morphism.h f
+    ; identity = refl
+    ; homomorphism = refl
+    ; F-resp-≈ = id
+    }
+
+  -- typedef
+  FreeUniformIterationAlgebra : Obj → Set (suc o ⊔ suc ℓ ⊔ suc e)
+  FreeUniformIterationAlgebra X = FreeObject {C = C} {D = Uniform-Iteration-Algebras D} forgetfulF X
+
+```
+
+### *Lemma 16*: definition of monad ***K***
+
+```agda
+  record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
+    field
+      algebras : ∀ X → FreeUniformIterationAlgebra X
+
+    freeF : Functor C (Uniform-Iteration-Algebras D)
+    freeF = FO⇒Functor forgetfulF algebras
+    
+    adjoint : freeF ⊣ forgetfulF
+    adjoint = FO⇒LAdj forgetfulF algebras
+
+    K : Monad C
+    K = adjoint⇒monad adjoint
+
+    -- EilenbergMoore⇒UniformIterationAlgebras : StrongEquivalence (EilenbergMoore K) (Uniform-Iteration-Algebras D)
+    -- EilenbergMoore⇒UniformIterationAlgebras = {!   !}
+```